Download Experiment No : M8 Experiment Name: FREE FALL and ATWOOD`S

Experiment No : M8
Experiment Name: FREE FALL and ATWOOD’S MACHINE
Purpose of the experiment: Determining the gravitational acceleration using a free
falling object and an Atwood’s machine. Investigating the position vs. time, velocity vs. time
relations for the two systems.
Theoretical background:
Gravitational acceleration and free fall
Newton’s law of universal gravitation dictates that two point particles placed at a distance will
attract each other with a force that is proportional to their masses and inversely proportional to
the square of the distance between them. The law also states that this force lies on the line that
connects two points and is directed towards the other body.
Its magnitude can be expressed by the following equation:
9.1
Here m1 and m2 represents the masses of point particles while r represents the distance
between them. G is called the universal gravitational constant.
Newton has also proved a theorem which states that a thin spherical shell of uniform mass
distribution gravitationally interacts with its outer region like a point particle as if its total
mass is concentrated at its center. (The second part of the theorem states that the shell doesn’t
interact with its interior region at all because of the cancellation of the forces.) This theorem is
called the spherical shell theorem. According to one rumor Newton waited several years to
publish his gravitational law because he wanted to find a legitimate proof to this theorem.
Now let us consider the motion of bodies on Earth’s surface using Newton’s gravitational law
and the spherical shell theorem. The objects that we encounter everyday are very small
compared to the Earth’s size so they can be treated as point particles with respect to Earth.
Furthermore, the spherical shell theorem allows us to treat the Earth itself as a point particle
given that its approximate shape is close to a sphere. So it is very reasonable to use equation
9.1 directly. The gravitational force on a body near Earth’s surface can then be calculated by:
9.2
Here M represents the mass of the Earth, R stands for its radius and m is the mass of the body.
If the only force on the body is the gravitational force then we can substitute this to Newton’s
second law of motion and write the equation of the motion as follows:
9.3
In this equation while m represents the body’s gravitational mass,
stands for its inertial
mass. If we use the equivalence principle which states that these two quantities can be
regarded as equal we can cancel out the mass of the body and we get the following expression
9.4 for the gravitational acceleration:
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9.4
As one can see the gravitational acceleration near Earth’s surface depends on the mass and the
radius of Earth. If we substitute the SI values of G, M and R to equation 9.4 we get a value of
9,80 m/s2 for the gravitational acceleration. Note that this quantity (which is generally
represented by the symbol g) is an approximate value and its exact value depends on the local
features (mainly latitude and altitude) of the geographical position on Earth’s surface.
Equation 9.5 is called the International Gravity Formula and can be used to estimate the value
of gravitational acceleration at a given geographical position on Earth.
g = 9,780327× (1 + A sin2 L - B sin2 2L) - 3.086×10-6×H
9.5
A = 0,0053024
B = 0,0000058
L = Latitude
H = Altitude (from sea level, in units of meters)
In this experiment you will study the motion of a free falling body by measuring the height
and the time of the free fall. Since it is a motion with constant acceleration the relationship
between those two quantities will be given by the kinematics equations that were derived in
the theoretical background of experiment 2. In the experiment there will be no initial velocity
so the relation of falling height and time can be written as:
9.6
Atwood’s Machine
Figure 9.1 Atwood’s machine
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Atwood machine is an apparatus invented by English mathematician George Atwood in the
year 1784 to study and verify the laws of motion with constant acceleration. It consists of two
bodies that move in the horizontal direction which are attached to each other by a string going
over a pulley as shown in Figure 9.1. To calculate the accelerations of m1 and m2 we begin by
drawing the free body diagrams of two bodies and the pulley as shown in Figure 9.2. m2 is
assumed to be greater than m1.
Figure 9.2 Free body diagrams of m1, m2 and the pulley
The equations of motion of the parts of Atwood’s machine can be written as follows:
9.7
9.8
9.9
While equation 9.7 and equation 9.8 are directly written from Newton’s second law of motion
equation 9.9 is the application of that law to the rotational motion of the pulley. While the
right hand side of Eq. 9.9 represents the net torque acting on the pulley, I term on the left hand
side stands for the moment of inertia and for the angular acceleration of the pulley. We have
3 equations but 4 unknowns which are T1, T2, and . So we need another equation to solve
the unknowns. The fourth equation comes from the observation (or hypothesis) that the rope
doesn’t “slide” when passing through the pulley which assures that the magnitude of the
tangential velocity of the pulley’s edge matches with the velocity of the objects. The
following relation between the angular and linear accelerations can also be directly derived
from this condition since the derivative of the velocities would also equate to each other.
9.10
Now given that we have four equations we can pull out T1 from Eq. 9.7, T2 from Eq. 9.8,
from Eq. 9.10 and substitute them in Eq. 9.9 to get the following identity:
9.11
can be solved from Eq. 9.11 as follows:
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9.12
Since the pulley has a shape of a disk its moment of inertia would be given by
. If we
substitute this in Eq. 9.12 we get the following expression for the linear acceleration of the
system:
9.13
As one can see from Eq. 9.13, the system will exhibit a motion with constant acceleration.
The kinematics equations of such a motion is given as follows: (for a comprehensive
derivation see the theoretical background of Experiment 2)
9.14
9.15
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Experimental Procedure
:
Figure 9.3 Free fall and Atwood machine experiment set-up
1. Experimental set-up is shown in Figure 9.3
2. Examine carefully the pulley, two sensors, mass holder and the bucket which are all fixed
to the main vertical rod. The uppermost sensor and the holder are fixed to the same metal
frame. The holder has an electromagnet inside which is always active when the main power of
system is turned on. When one fixes a metal body to this holder the timer automatically resets
itself. When the button is pushed the electricity feeding the electromagnet of the holder is cut
and the body is released. Meanwhile the timer starts measuring the time at the moment of
release and stops when the body passes through the second sensor which is further down on
the main rod. The bucket with the sponge inside serves to stop the falling bodies gently
without causing damage to the apparatus.
3. The metal frames which the sensors and holder are fixed. They can be moved up and
down by loosening from the small black arms on their right. Make sure you are holding the
frame with your other hand as you turn the arm to loosen the frame. The heights can be read
from the ruler that is fixed behind the main rod. There is a white line on the frame to make the
reading easier. Always record your measurements in millimeter accuracy.
4. The experiment consists of two parts, namely free fall and Atwood’s machine.
5. Before performing the Atwood machine experiment align the holder height such that the
masses on both sides should be able to swing freely without touching anywhere even one of
the masses is at its topmost position. Don’t forget to stop the swing motions of the masses by
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your hand before the measurement. To avoid unnecessary friction make sure that the rope
touches nowhere other than the pulley during the experiment. Ask for help from the
experiment responsible if necessary.
6. Before performing free fall experiment remove the rope from the pulley and remove the
masses away from the falling path of the object. Take one of the metal balls and stick it to the
holder. Make a trial release by pushing the button to make sure that the ball falls in the
bucket. Record the holder height to the part on top of Table 9.1. To cope with the random
errors you are asked to perform 5 measurements for 5 different heights. You can take an
approximate value of 20 cm between your release heights. Always record the exact value that
you read from the ruler.
7. Record your values to Table 9.1.
Holder height: .......................................
Table 9.1 Height vs. time table for free fall experiment
Sensor Height
(cm)
t1
t2
Time (s)
t3
t4
t5
8. Remove the metal ball and pass the rope over the pulley for Atwood machine experiment.
(Read step 5 again) Ask the experiment responsible the values of masses to hang on both
sides.
9. Take measurements in the same manner as the free fall experiment. (5 time
measurements for 5 different heights.) Record your values to Table 9.2
Holder height: .......................................
m1 = ........................
m2 = ............................
Table 9.2 Height vs. time table for Atwood machine experiment
Sensor Height
(cm)
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t1
t2
Time (s)
t3
t4
t5
Analysis and Graphs
You are asked to plot height vs. time and velocity vs. time graphs for free fall and Atwood
machine experiments.
Height vs. time Plots
Fill in the right coumn of Table 9.3 by subtracting the heights in Table 9.1 from the holder
height. Fill in the left column of Table 9.3 by the average values of the 5 time measurements
of the corresponding heights.
Table 9.3 Fall time vs. fall height table for free fall experiment
Fall time (…..)
Fall height (…...)
Express the data in Table 9.3 as points on your plot. Regarding the theoretical background,
what type of a curve is expected to pass through the points?
…………………………………………………………………………………………………
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…………………………………………………………………………………………………
Draw this curve by fitting it to points as good as you can by your crude eye estimation.
Fill in the right coumn of Table 9.4 by subtracting the heights in Table 9.2 from the holder
height. Fill in the left column of Table 9.4 by the average values of the 5 time measurements
of the corresponding heights.
Table 9.4 Fall time vs. fall height table for Atwood machine experiment
Fall time (…..)
Fall height (…...)
Express the data in Table 9.3 as points on your plot. Regarding the theoretical background,
what type of a curve is expected to pass through the points?
…………………………………………………………………………………………………
…………………………………………………………………………………………………
…………………………………………………………………………………………………
Draw this curve by fitting it to points as good as you can by your crude eye estimation.
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Velocity vs. time plots and gravitational acceleration calculation
The velocities are not directly measured in the experiment hence you need to calculate them.
If we consider the relation between height and the time
and move one of the t’s on
the right hand side to the left we get the following equation:
9.16
The left hand side of 9.16 has the dimension of velocity and equates to the average velocity of
a freely falling body from a height of h. To be able to plot this value as an instantaneous
velocity (by a point on graph) it should be plotted against the half of the flight time as can be
seen from the equation. Give a physical explanation of this below:
…………………………………………………………………………………………………
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Fill in the left column of Table 9.5 using the half of the time values in the Table 9.3. Fill in
the right column of Table 9.5 by dividing the height values to the time values in the Table 9.3.
Table 9.5 Velocity vs. time table for the free fall experiment
Time (…..)
Velocity (.....…...)
Express these data as points on your plot. Since it is expected to be a motion with constant
acceleration we expect a line with the equation
to pass through these points. You are
asked to calculate the gravitational acceleration g which corresponds to the slope of this line.
Use the linear fitting formula to calculate g below:
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………………………….
Use this value to plot the line on your graph and observe how it fits with your experimental
points.
Use Eq. 9.5 to calculate the gravitational acceleration at the lab that you performed the
experiment.
(Gebze Technical University latitude: North 40,81o, Altitude: 13 meter)
……………………….
Compare the two values and calculate the percentage difference. Discuss the possible reasons
for the differences.
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Now we are going to repeat the same procedure for Atwood machine experiment. But this
time we expect the acceleration to be given by Eq. 9.13. Using the same logic use the values
in Table 9.4 and the equation
to fill in Table 9.6.
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Table 9.6 Time vs. velocity table for the Atwood machine
Time (…..)
Velocity (.....…...)
Express these data as points on your plot. We expect a line with the equation
to pass
through these points. You are asked to calculate the acceleration of the Atwood machine
which corresponds to the slope of this line. Use the linear fitting formula using the values of
Table 9.6 to calculate the acceleration below:
Draw the line on your graph using this acceleration and observe the fitness of the line to data
points.
Use Eq. 9.13 to calculate the expected value of the acceleration. The mass of the pulley is 8
grams. (M = 8 gr).
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Compare the two values with each other. Calculate the percentage difference and discuss the
possible reasons for this difference.
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Calculate
using
. What is the percentage error? Which of the experiments
(free fall or Atwood’s machine) has less error? Discuss the reasons.
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Questions:
1. Use Eq. 9.5 to calculate how much the gravitational acceleration differs on the poles
and the equator of the Earth at sea level.
2. Compare the distribution of consecutive time values measured at same height in the
free fall and Atwood machine experiment. Are the random errors same in these
experiment. How can this be expressed mathematically. If the random errors are
different what might be causing the difference?
3. Watch the video named: “Weak Equivalence Principle test on the moon” at the link:
https://www.youtube.com/watch?v=MJyUDpm9Kvk.Calculate the falling times of the
feather and the hammer which the astronaut drops. (Use Eq. 9.4 to calculate the
gravitational acceleration on moon, investigate the other necessary constants on
internet, assume he is dropping from a height of 1,5 meters) Does this time matches
the one that you observe in the video?
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